Uncertainty prediction is a challenging topic, important in data assimilation as well as in probabilistic forecasting. One of the main theoretical backbone is given by the Kalman filter equations that applies for linear dynamics, but that fails to apply for large system where the numerical cost to predict the error covariance becomes prohibitive. Hence, approximations of the KF have been proposed, as the ensemble Kalman filter (EnKF) where the error covariance matrices are approximated from ensemble estimation [1]} ; or recently the parametric Kalman filter (PKF) where the error covariance matrices are approximated from a covariance model [2]}. In the PKF, the dynamics of the parameters provides a proxy for the dynamics of the full covariance matrix. For instance, covariance model parameterized from the variance and the anisotropy of the local correlation functions are able to predict the dynamics of the covariance matrix for transport equations [3]}, but at a numerical cost equivalent to three time the integration of the transport [4]}, [5]}. In addition, the PKF provides a view of the dynamics of the uncertainties that cannot be understood from an ensemble estimate alone. And this has open new ways to tackle difficult topics e.g. the dynamics of the model-error covariance [6]}, [7]}. Note that other insight in theoretical covariance dynamics have been recently proposed [8]}.
